Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{p^2 - 25}{p + 5}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = p$ $ b = \sqrt{25} = 5$ So we can rewrite the expression as: $q = \dfrac{({p} + {5})({p} {-5})} {p + 5} $ We can divide the numerator and denominator by $(p + 5)$ on condition that $p \neq -5$ Therefore $q = p - 5; p \neq -5$